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Results tagged with ap.analysis-of-pdes
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user 66623
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
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answer
140
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Hardy inequality
Suppose $B_1$ is the unit ball centered at the origin in $ R^N$ with $N \ge 3$. Let $ q= 2^* = \frac{2N}{N-2}$. Does there exist some $C>0$ such that
$$\int_{B_1} \frac{ u(x)^2}{|x|^2} dx \le C \| …
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53
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Estimating a potential
Let $ \Omega$ denote a smooth bounded domain in $R^N$ where $N \ge 3$ and we let $u \in C^3( \overline{\Omega})$. Let $ \delta(x) = \operatorname{dist}(x, \partial \Omega)$. For $ x \in \Omega$ (but …
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Reference request; fractional Laplacian; boundary regularity
Consider $B_2^+$ the half ball in $R^N$ and consider
$ (-\Delta)^s u = f(x) $ in $B_2^+$ with $ u=0$ outside. Is there any references where someone tries to use an odd extension of $u$ across $ x_N …
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54
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Sign changing elliptic problem
Take $B_1$ the unit ball in Euclidean $N$ dimensional space and suppose $3 \le N \le 10$ and take $ 1<p< \frac{N+2}{N-2}$. By some abstract theory there is a infinite sequence of smooth radial sign …
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Perturbation of first eigenvalue
Let $\Omega$ be a bounded domain in $R^N$ and suppose $\gamma$ is some smooth bounded function in $\Omega$. Let $ -\Delta \phi_0 = \lambda_0 \phi_0 $ in $ \Omega$ with $ \phi_0=0$ on $ \partial …
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4
votes
1
answer
163
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Smoothness of critical elliptic problem
I am convinced I have seen results along the lines of: if $ u \ge 0$ is an $H_0^1(\Omega)$ solution of
$$-\Delta u = u^{q-1}$$ in $\Omega$ with $ u=0$ on $ \partial \Omega$ (here $\Omega$ is a smooth …
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2
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97
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A log cut off function
Let $\Omega$ denote a bounded smooth domain in $R^N$ and consider $\Gamma$ a smooth subset (assume its some $k$ dimensional manifold where $k \le N-1$). Let $ \delta(x)=dist(x, \Gamma)$. On occa …
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63
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Liouville theorem for linear equation
Let $\Omega$ denote a an exterior domain in $R^N$ with smooth boundary.
I am interested in Liouville Theorems related to smooth solutions of
$$\Delta \phi(x) + \gamma \sum_{i,j=1}^N \frac{x_i x_j}{|x| …
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105
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Gradient bounds on a solution of a linear elliptic problem
Take $\Omega$ to be a bounded domain in $N$ dimensional Euclidean space with smooth boundary and we assume $\Omega$ contains the origin. I am interested is the following equation
$$ \Delta \phi(x) …
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129
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Sturm-Liouville result
Suppose $n \ge 2$ an integer and consider finding the first eigenvalue of
$$ -\partial_\theta \left( \omega(\theta) \psi'(\theta) \right) = \mu_1 \omega(\theta) \psi(\theta)$$ for $ 0<\theta<\frac{\pi …
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156
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barrier functions ; harmonic functions; gradient estimates (textbook reference)
Suppose $\Omega$ is a bounded domain in Euclidean space and you can take it as convex as you wish. Is there a good place to look for a bunch of barrier type functions.
In particular I am looking for …
- 1,385
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answers
47
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Odd extension of radial solutions of elliptic pde
I have a question about trying 'odd' like extension to obtain some sign changing solutions of an elliptic equation.
Lets first consider the 1 dimensional problem. To solve a sign changing solution of …
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3
votes
1
answer
133
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Positive first eigenvalue; operator satisfies maximum principle
I am attempting to understand a paper. They have $u$ is a stable smooth solution of $ -\Delta u = f(u)$ in $ \Omega$ with $ u=0$ on $ \partial \Omega$ where $\Omega$ is a bounded domain in Euclidean …
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Elliptic pde and boundary layer estimates
I have a question that is related to finding some boundary layer estimates. I am sure there is a general method for this but I don't know it.
I will pose the problem in two dimensions (but here there …
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50
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Rescaling an elliptic system
I have a basic question regarding rescaling an elliptic system when trying to get apriori estimates.
Consider an elliptic system say of the form
$$-\Delta u(x) = u^{p_1} v^{q_1}, \quad -\Delta v = u^{ …
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